Energies 2012, 5, 4779-4803; doi:10.3390/en5114779

energies ISSN 1996-1073

www.mdpi.com/journal/energies

Article

Non Breaking Wave Forces at the Front Face of Seawave Slotcone Generators

Mariano Buccino 1,*, Davide Banfi 1, Diego Vicinanza 2, Mario Calabrese 1,

Giuseppe Del Giudice 1 and Armando Carravetta 1

1 Department of Hydraulic, Geotechnical and Environmental Engineering, University of Naples

Federico II, Via Clauio 21, Naples, Italy; E-Mails: davidebanfi@hotmail.com (D.B.);

calabres@unina.it (M.C.); Giuseppe.delgiudice@unina.it (G.D.G.);

Armando.carravetta@unina.it (A.C.) 2 Department of Civil Engineering, Second University of Naples, Via Roma 29, 81031 Aversa

(Caserta), Italy; E-Mail: diego.vicinanza@unina2.it

* Author to whom correspondence should be addressed; E-Mail: buccino@unina.it;

Tel.: +39-0817683428; Fax: +39-5938936.

Received: 17 September 2012; in revised form: 1 November 2012 / Accepted: 16 November 2012 /

Published: 19 November 2012

Abstract: The Seawave Slotcone Generator (WAVEnergy SAS, 2003) is a wave energy

converter based on the overtopping principle. Although it has been effectively researched

during the last decade, no design tool has been supplied to estimate the hydrodynamic

loads the waves exert on its front face. In this article a set of well reliable 3D experiments

has been re-analyzed, in order to get indications on possible calculation methods. It is

shown that the Japanese design tools for monolithic sea dikes may be reasonably adapted

to the present case. Finally a new approach is presented, which is based on the so called

momentum flux principle; the resulting predictive equation fits the experimental data

remarkably well.

Keywords: wave energy converters; Seawave Slotcone Generator (SSG); wave forces;

physical model tests

OPEN ACCESS

Energies 2012, 5 4780

1. Introduction

The Seawave Slotcone Generator (SSG, Figure 1) is a Wave Energy Converter (WEC) based on the

overtopping principle (see [1]). The wedge of water that climbs the front-face during the up-rush phase

is captured in a number of reservoirs (generally two or three) placed on top of each other; thus, the

water, on its way back to the sea, passes through a turbine, spinning it and generating electricity. A

number of studies [2–4] suggest that the use of multiple reservoirs results in a higher overall efficiency

compared to structures which employ a single tank.

Figure 1. Artistic representation of a 3-level SSG (WAVEnergy SAS, 2003).

SSG devices have been well researched during the last decade and two pilot plants have been

planned to be located along the west Norwegian coasts; one at the isle of Kvitsøy (in the Bokna fjord,

not far from Stavanger, Figure 2) and at the other at Svåheia (very close to Kvitsøy). Although these

projects have not been realized, they have had however the result of intensifying the research activity.

Indeed, the latter has also been encouraged by the fact that both the structure of the device (concrete

monolithic dike) and its working principle (run up, overtopping) are well known to researchers in the

field of maritime engineering.

Figure 2. The Kvitsøy site.

Energies 2012, 5 4781

A recent paper published on this journal [5] reviews the fruitful work done so far. Yet, going

through the pages of the article, it becomes clear that many design tools have been developed for

various aspects of the technology (e.g., for predicting the mean overtopping rate, the hydraulic and

overall efficiencies, the wave reflection, etc.), except for the structural response. In fact the tests

performed by Vicinanza and Frigaard [6] to estimate the wave loadings acting on the Kvitsøy pilot

system did not lead to a real design formula, but only to some order of magnitude indications.

However, the authors noted that some of the design tools traditionally employed by coastal engineers

could result in significant underpredictions of the pressure peaks.

This lack of knowledge may be central to the future of this technology, because the SSG is by

nature exposed to severe seas and the economic impact of the structural design could be so heavy as to

discourage investors. In this regard, it is obvious that the want of clear indications and the absence of

ad hoc experimental analyses, drive engineers to build by themselves design criteria that will be

invariably too safe and expensive. The scope of this paper is to start to fill that gap.

The experiments of Vicinanza and Frigaard are re-analyzed here in order to suggest possible

calculation methods. The starting point of our work is the comparison with the Japanese design tools

for monolithic sea defense structures. Then a new approach is presented, which proved to work

remarkably well with reference to the available data.

2. Experiments

The experiments have been carried out at the Department of Civil Engineering of the University of

Aalborg, Denmark [6].

2.1. Tests Description

The tests have been performed in a basin 15.7 m long, 8.7 m wide and 1.5 m deep provided with 10

independent piston type paddles, capable of generating both long crested (2D) and short crested (3D)

waves. Both the SSG and the bathymetry of the site have been reproduced in the tank at a 1:60

length-scale ratio. The SSG device has been located at the top of a steep rocky cliff, inclined 1:1 with

respect to the horizontal (Figure 3).

Figure 3. Cross section of the bathymetry.

DEVICE

ROCKY CLIFF ( 1 : 1 )

PLATEAU

h = 0.50 mh' = 0.53 m

Energies 2012, 5 4782

The role of the cliff is enhancing the run-up height; in the following we’ll refer to it as “run-up

ramp” or “focuser”. Seawards of the cliff, the bottom has been kept constant for 5 m, corresponding to

300 m in the prototype. The SSG model, made of Plexiglass, included three reservoirs and three front

plates with a 35 degrees front slope angle (Figure 4).

Figure 4. Sketch of the SSG model (dimensions are in mm).

Altogether 32 JONSWAP driven random sea states were run, 16 with long crested waves (2D) and

16 with short crested waves (3D). The spectral significant wave height, Hs, varied from 0.042 m to

0.25 m and the peak period, Tp, ranged from 1.03 s to 2.07 s.

2D tests have been conducted using two different water levels at the toe of the cliff (h), and namely

0.53 m (eight tests) and 0.5 m (eight tests). The former corresponds to a water depth at the toe of the

device (d) equal to 0.03 m (1.8 m in prototype), while the latter leaves the SSG completely exposed

(shoreline device, d = 0 m). For each array of eight tests, four tests have been carried out with normal

waves (“Front waves”, F) and four with oblique waves with an angle of propagation relative to the

normal to the structure (β0) equal to 45 degrees (“Side waves”, S). Short crested waves have been

generated using a cosine power directional spreading function:

25

24

2322

21

20

18

17

19

471

97

67

70

76

17

9

35°

3° RESERVOIR

2° RESERVOIR

1° RESERVOIR

Energies 2012, 5 4783

; cos | | with2 2

(1)

where β is the direction of propagation of the generic Fourier component and β0 is the mean direction

of the waves. n is a spreading index; the larger is n, the lower the dispersion of the spectral

components. 3D tests have been conducted with a single value of h, i.e., 0.53 m. The same array of

eight tests used for the long crested waves has been repeated twice with two different values of n,

namely n = 10 (small spreading) and n = 4 (large spreading).

The wave pressures at the structure have been recorded through 14 pressure cells sampled at 200 Hz.

Nine of them were located on the front face, three for each plate.

2.2. Breaker Types and Characteristics of Loadings

Vicinanza and Frigaard observed that at the front face of the SSG, the waves took the shape of

surging breakers, which climbed the structure inducing a massive overtopping. The occurrence of

surging breakers, which in fact are non-breaking waves, produces pressures with of a relatively low

magnitude (on the order of Hs in the water column) and a slow variation in time. This can be easily

observed from Figure 7 of the Vicinanza and Frigaard paper. However, it has been also noted that

under oblique wave attacks a part of the wave front violently hit the side wall of the SSG, inducing a

rapidly varying intense loading.

2.3. Model and Scale Effects

Before performing the analysis, it could be of interest to briefly highlight the possible source(s) of

bias between the model results and the prototype reality. As reasoned above, the absence of a real

breaking at the front face of the structure produced slowly acting loads, generally termed “pulsating”

or “quasi static”; it is generally well accepted that this kind of forces are free of significant scale

effects [7]. Obviously, this is true so long as all the phenomena which govern the wave structure-

interaction are well reproduced in the model. In the present context, the most relevant interaction mode

is obviously wave overtopping. In this regard, it has long been known that a possible source of

distortion comes from surface tension effects, which, however, have been recognized to be relevant

only for very shallow water depths (h < 0.02 m) and/or for very short periods (T < 0.33 s) [8].

Unfortunately, most recent investigations conducted within the research project CLASH [9], funded by

EU, proved model and scale effects on wave overtopping are very hard to avoid, being the mean

overtopping discharge in prototype larger than in the model; yet the difference reduces with increasing

overtopping rate.

Since in most of present experiments the amount of water overpassing the structure crest was huge,

it is expected that the bias to be small; furthermore, as wave loads acting on the wall reduce with

increasing overtopping, it can be argued that model results presented below might be slightly

conservative compared to the actual prototype conditions.

More significant scale effects are expected to affect the breaking wave pressures at the side walls of

the SSG. A number of literature studies have indicated that, for plunging waves, the real wave pressure

may be largely overestimated by up to 60% or more by Froude scaling. An interesting method to

evaluate the magnitude of the bias has been recently proposed by Cuomo et al. [10], based on the

Energies 2012, 5 4784

so-called Bagnold number. The features of the wave actions at the side walls are not dealt with in this

paper, but represent an intriguing topic to be researched in the future.

Before concluding this discussion, a further aspect deserves to be addressed. As the experiments

were conducted in a rectangular basin, the incoming waves may excite the natural oscillation modes of

the facility, giving rise to resonance phenomena. This would produce parasitic standing waves in the

tank, that may affect the reliability of the results. However the simulated sea-states have very short

frequency bands, included between 0.25 and 2 Hz, so that the excited frequencies are short enough to

be absorbed by the rubble beach placed at the end of the basin and to be damped out by the friction of

the concrete walls of the tank. This reasoning was also supported by the good agreement between the

wave spectra generated by the wavemaker and those measured by a set of seven resistive probes spread

throughout the basin [6].

3. Japanese Design Practice for Monolithic Maritime Dikes

As briefly mentioned before, a possible point of strength of the SSG may be its similitude with

structures usually employed in the field of maritime engineering. This property implies a certain

degree of “a priori” knowledge about the processes that govern the wave-structure interaction, which

might put the SSG technology a step forward compared to other types of WEC.

As an example, the formula for calculating the mean overtopping rate at the various reservoirs [11]

has the same structure as that commonly used for the rubble mound breakwaters [12] and the

procedure for evaluating the efficiency of the device includes an algorithm to estimate the “wave by

wave” overtopping discharge, which has been originally suggested for vertical breakwaters [13].

As far as the structural response is concerned, the reference is of course that of the monolithic

maritime dikes, including vertical face breakwaters [Figure 5(a)], sloping top caissons [Figure 5(b)]

and trapezoidal walls [Figure 5(c)].

Figure 5. Examples of monolithic maritime dikes: (a) Vertical breakwater. (b) Sloping top

caisson. (c) Trapezoidal wall.

- 13,7

+ 5,7

- 0+

+ 8,8

Dover breakwater

a b

c

Energies 2012, 5 4785

In this paper, particular attention has been drawn to the formulae of the Japanese design practice;

this because they are calibrated to work on structures having two features in common with the SSG,

namely:

1. a water depth at their toe relatively small;

2. a low crest, allowing for a massive overtopping under the design wave conditions.

Although the methods employed in the following are widely known, a synthetic review is provided

in the paragraphs below, both for sake of clarity and to facilitate practical applications.

3.1. Vertical Face Breakwaters: the Formulae of Hiroi and Goda

During the design of the Otaru Port breakwater (1912 through 1917) Hiroi conducted in-situ

measurements of wave pressures by means of a spring-type instrument. Assimilating the pressure

exerted by the waves to that of a jet impinging a plane surface, the author came to the formula:

P = 1.5 ρgH (2)

where ρ is the water density, g is the gravity acceleration and H is the design wave height. The wave

pressure P is assumed to act uniformly up to the minimum between the crest of the structure and

1.25 H (Figure 6).

Figure 6. Definition sketch for the Hiroi formula.

Despite the fact that the Hiroi formula is “apparently intended for calculating the pressure caused

by breaking waves” [14], it has been however considered, both in virtue of its simplicity and because

of the similitude with an impinging jet, physically appropriate to the case of the SSG.

In 1973 Goda proposed his worldwide-known formula, based on the results of the theoretical and

experimental investigations the author conducted between the late 60s and the early 70s. One of the

most immediate advantages of the Goda method is that it holds for all the wave conditions, i.e., for

both standing and breaking waves. As shown in Figure 7, the pressure distribution under the wave

crest is generally bi-trapeizoidal , with a zero at the run-up height. The latter is calculated as:

hc

d

h

Energies 2012, 5 4786

∗ 0.75 1 cos (3)

in which β0, as stated above, is the angle of the wave propagation relative to the normal to the

breakwater.

At the mean water level, the wave pressure p1 is assumed to be linear in the wave height, although

the author noted that “the relative pressure intensity varies with the increase in the relative wave

height to a certain extent…” [14]. The expression of p1 is:

0.5 1 cos cos (4)

where the parameter α1 depends only on the water depth to wavelength ratio h/L:

0.6 0.5

4

senh 4

(5)

and increases from 0.5 in deep water (h/L ≥ 0.5) to 1.1 in shallow waters (h/L ≤ 0.05).

The coefficient α2 accounts partially for the effects of impulsive loadings and vanishes in the

absence of a rubble mound foundation. Its original expression given by Goda was successively

modified by Takahashi et al. [15], with the purpose of improving the performance of the formula under

plunging breaker conditions.

The wave pressure at the bottom, p2, is calculated using the transfer function of the linear waves:

cosh (6)

so that the pressure exerted at the toe of the structure, p3, can be calculated by linear interpolation.

Figure 7. Definition sketch for the Goda formula.

p1 hc

d

h

p3

p2

Energies 2012, 5 4787

3.2. Sloping Top Caissons

When the breakwaters have to face very rough seas, their upper part may be inclined to reduce the

horizontal thrust; in addition, the vertical downwards component of the wave force exerted on the

sloping part tends to compensate for the uplift force at the breakwater basement, with the result of

improving the resistance against overturning. As noted by Takahashi et al. [16], the first sloping top

breakwater was constructed in 1906 to defend the Port of Naples (Italy); in Japan the demand for such

structures had been growing for some years in the mid-90s, requiring the development of ad-hoc

procedures for their design.

The starting point of the Japanese approach to the sloping top caissons is the similitude with the

hydrodynamic thrust exerted by a jet of fluid over an inclined plane surface (Figure 8). As mentioned

above, the same criterion had inspired the Hiroi formula.

If we indicate with MF the momentum flux of the incoming jet, then, under the hypothesis that after

the collision with the slope the velocity of fluid is exclusively tangential to the structure, one gets:

sin (7)

in which is the angle of the front face to the horizontal.

Figure 8. A fluid jet hitting an inclined wall.

In the method by Morihira and Kunita [17], the momentum flux comes from the Goda’s pressure

distribution (Figure 9) so that MF can be calculated integrating the loadings (along z) from the lower

tip of the sloping part (dc ) to the crest of the breakwater (hc):

(8)

Takahashi et al. [16] performed a number of regular wave experiments and found out that the

method above led to an underestimation of the forces for small waves and an overprediction of the

forces for high waves. Hence the authors proposed the introduction of an uniform correction factor λSL

to be applied to the Goda model in the sloping part of the breakwater. That is:

(9)

MF

MF

FP

Energies 2012, 5 4788

Figure 9. Definition sketch for the method of Morihira and Kunita.

The expression of is:

min max 1.0; 231

tan0.46

1tan

1sin

; 1

sin (10)

where H/L is the wave steepness. Note that the authors proposed a different coefficient, λV, for the

upright face of the structure (below dc).

3.3. Trapezoidal Walls

Tanimoto and Kimura [18] performed a series of laboratory experiments to investigate the

magnitude and the distribution of the wave loadings acting on a trapezoidal caisson (Figure 10). The

authors came to the conclusion that the Goda model could be simply rotated on the outer face of the

breakwater. The experiments also demonstrated that the uplift wave pressure on the caisson bottom is

reduced due to the upward water particle velocity enhanced by the slope.

Figure 10. The method of Tanimoto and Kimura.

Note that the Tanimoto and Kimura approach leads to values of wave force well larger than those

predicted by the Morihira and Kunita method discussed in the previous paragraph. This because the

F1p1 hc

dc

p3

p2

Rotation

Energies 2012, 5 4789

Goda’s pressure distribution is assumed to be normal to the wall (instead to be horizontal); in addition

it is integrated along the outer face of the structure (instead of its vertical projection, Figure 9).

3.4. Selection of Wave Parameters

The experiments used to carry out the design tools presented above, employed mostly regular

waves; hence the problem arises of how applying the formulae to real sea states. The approach used in

the modern Japanese practice, say from the late 60s [19], is that of assimilating a random sequence of

waves to a series of regular waves, assuming that the prediction methods are valid “wave by wave”.

Moreover the principle of breakwater design is essentially deterministic: the structure should withstand

the greatest force expected in its service life, which would be exerted (on the statistical average [14])

by the highest wave among a train of random waves corresponding to the design condition. In other

words, the maximum wave force acting onto the structure, say Fmax, has to be calculated by inserting in

the design formulae the maximum wave height of the design storm, Hmax, and its period Tmax. Being

aware of the stochastic nature of these quantities, Goda highlighted their numeric evaluation could be

achieved but in a conventional way. Thus, for waves which are stable on the sea bottom (non-breaking

waves) he proposed:

1.8 (11)

being Hs and Ts the significant wave height and period respectively. The definition of Hmax corresponds

to the average of the 1/250th highest waves in a narrow banded random sea state and for this reason it is

often indicated also as H1/250. Consequently, in laboratory and practical engineering applications, the

maximum wave force Fmax is generally intended as the average of the highest 1/250th peaks of force in a

given random sequence (F1/250). For breakwaters located in shallow waters Goda [19] provided a method

to calculate the maximum wave height taking into account the effects of the random wave breaking.

As far as the Hiroi formula, which dates back to the 1920s, is concerned, Japanese engineers were

used to applying it employing Hs and, as stated by Goda [14], “the formula had not betrayed their trust

in its reliability”.

4. Application to SSG

4.1. Definition of the Maximum Wave Force F1/250

In this study the time histories of the wave pressures at the different transducers were not available.

Thus the maximum wave force F1/250 has been conservatively calculated by integrating the values of

the maximum pressures p1/250 along the front face of the structure. This definition is cautious because,

due to the obliquity of the wall, the pressure signals at the different locations are not in phase.

However it should be noted that under non-breaking wave conditions the variation of loadings in the

time is quite slow and accordingly the phase lags are expected to be relatively small.

Energies 2012, 5 4790

4.2. How the Prediction Formulae Are Used

In the following the Tanimoto and Kimura approach for trapezoidal walls is used. The ensemble

“SSG + ramp” has been assimilated to a sloping structure and the Goda and Hiroi pressure

distributions have been rotated perpendicularly to them (Figure 10). As far as the Goda method is

concerned, it should be highlighted that no rubble mound foundation is supposed to exist;

consequently the coefficient α2 in Equation (4) is always zero.

Consistently with the Japanese practice, the significant wave height Hs has been used to calculate

the wave pressure with the Hiroi formula, whereas H1/250 = 1.8 Hs has been employed for Goda. As far

as the wave period is concerned, the peak period Tp has been used instead of Ts. This because in

practical applications, engineers have generally direct information on Tp, whereas Ts is derived from

the former by the introduction of an empirical coefficient; the latter is close to 1 and varies within a

given range (Goda suggested 0.90–0.95). Since the choice of such a coefficient represents a source of

uncertainty, it seemed more convenient to perform the comparison using directly Tp.

Vicinanza and Frigaard [6] showed that the method of Takahashi et al. [16] for sloping top caissons

led to an underestimation of the maximum wave pressures by up to 50%. Hence in this study it will be

used in a heuristic manner. Basically, after rotating the Goda distribution on the front face of the SSG,

the pressures are multiplied by the factor of Equation (10). Moreover, in the calculation of this

coefficient it has been assumed that the effect of the wave steepness on loadings is best represented by

the significant wave steepness Hs/Lp rather than H1/250/ Lp. In fact the latter leads to values too big,

which reduce the resulting force dramatically.

5. Results for Long Crested Waves

5.1. Goda and Takahashi et al.

In general neither the simple Goda formula nor the use of the Takahashi et al. correction [16] lead

to predict properly the magnitude of the measured pressures p1/250. This is displayed in Figures 11

and 12, which show the data are rather scattered around the line of perfect agreement, with a relative

error reaching and surpassing 50%.

Figure 11. Measured p1/250 vs. predictions of the Goda formula. Left panel: Front waves;

right panel: side waves. Data in kN/m2.

0.0

1.0

2.0

3.0

4.0

0.0 1.0 2.0 3.0 4.0

TEST 1

TEST 2

TEST 3

TEST 4

TEST 9

TEST 10

TEST 11

TEST 12

+ 50% Band

- 50% Band

meas.

Goda0.0

1.0

2.0

3.0

4.0

0.0 1.0 2.0 3.0 4.0

TEST 5

TEST 6

TEST 7

TEST 8

TEST 13

TEST 14

TEST 15

TEST 16

+ 50% Band

- 50% Band

meas.

Goda

Energies 2012, 5 4791

Figure 12. Effect of Takahashi et al. correction [Equation (10)]. Left panel: Front waves;

low panel: right waves. Data in kN/m2.

Yet, a deeper insight into the results reveals that, for any given wave attack, both formulae return a

reasonable estimate of the mean pressure along the wall (pm250) and consequently of total thrust acting

on the front face of the SSG. From Figure 13 it is clear the relative error on pm250 is generally less than

20%, which may be considered acceptable from an engineering point of view. It is of interest that with

the adjustments proposed in this paper, the Takahashi et al. method does not produce the severe

underestimations reported by Vicinanza and Frigaard [6].

Figure 13. Comparison between measured and predicted values of the mean pressures

along the wall. In abscissa the test code.

Figure 14 indicates that the overturning moment about the shoreward toe (heel) of the structure is

also reasonably predicted by the formulae. Note all the values are negative, meaning that the moments

are opposing to the overturning of the SSG. This is due to the fact that the front face of the structure is

inclined more than 45° to the vertical, so that the vertical component of the wave force, which has a

stabilizing effect, prevails over the horizontal one.

From the foregoing discussion one may come to the conclusion that despite an unsatisfactory

estimate of the local values of the pressures, the Goda and Takahashi et al. methods can acceptably

0.0

1.0

2.0

3.0

4.0

0.0 1.0 2.0 3.0 4.0

TEST 1

TEST 2

TEST 3

TEST 4

TEST 9

TEST 10

TEST 11

TEST 12

+ 50% Band

- 50% Band

meas.

Tahakashi0.0

1.0

2.0

3.0

4.0

0.0 1.0 2.0 3.0 4.0

TEST 1

TEST 2

TEST 3

TEST 4

TEST 9

TEST 10

TEST 11

TEST 12

+ 50% Band

- 50% Band

meas.

Tahakashi

0

0.4

0.8

1.2

1.6

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Goda F

Takahashi F

Goda S

Takahashi S

TEST # 

pm250,calc. / pm250,meas.

Energies 2012, 5 4792

predict both the value of the total wave thrust and that of its arm. This is a remarkable result, since

SSG is a rigid structure, the failure of which does not depend on the spatial distribution of wave

loadings, but only on their resultant.

Figure 14. Measured vs. predicted torques at the structure heel. Left panel: Goda model;

right panel: effect of Takahashi et al. correction.

However, the formulae seem to suffer from some lack of fit, which can be best discussed with the

aid of the Figure 15. Here, the non dimensional values of the measured mean pressure, pm250/γHs, are

plotted versus the wave height to depth ratio Hs/h. The points are divided into a number of groups

depending on the local wave steepness, Hs/Lp. The parameter Hs/h has been selected to favor the

comparison with the findings of Takahashi et al. [16], who employed that index as starting point of

their analysis.

Figure 15. Non dimensional values of the mean pressure as function of the wave height to

depth ratio.

The solid curves in the graph represent the mean trends of the formulae of Goda (red curve) and

Takahashi et al. (blue curve). They have been drawn fitting the predicted values of pm250/γHs with

polynomials of different orders, up to get the maximum determination index R2.

‐140

‐120

‐100

‐80

‐60

‐40

‐20

0

‐140‐120‐100‐80‐60‐40‐200

Front

Side

Perfect Agreement

+20% Band

-20% Band

Goda  [Nm/m]

Mmeas.

[Nm/m]

‐120

‐100

‐80

‐60

‐40

‐20

0

‐120‐100‐80‐60‐40‐200

Front

Side

Perfect Agreement

+20% Band

-20% Band

Tahakashi  [Nm/m]

Mmeas.

[Nm/m]

0.0

0.3

0.6

0.9

1.2

1.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6

s 0.042

s 0.046

s 0.053

s 0.059

s 0.026

s 0.036

Goda's mean trend

Takahashi's mean trend

Hs /h

pm250/γHs

Energies 2012, 5 4793

Examining the figure, we note the experimental data exhibit an increasing trend, for Hs/h ≤ 0.3,

followed by a slightly decreasing one. On the contrary the Goda formula has a monotonic shape which

tends to underpredict data with low Hs/h (apart from a single outlier circled in the graph) and to

overpredict data with high Hs/h. The threshold between underestimates and overestimates seems to be

about 0.30. All these observations are absolutely consistent with the results of Takahashi et al.

On the other hand, the method of Takahashi et al. reproduces the data trend better; this is not

surprising, since the authors developed their formula just to correct the bias of the Goda model

observed in Figure 16. However there’s still a tendency to underpredict data with Hs/h ≥ 0.3, which is a

very common situation under design conditions. This aspect deserves to be carefully studied in more

depth; in the meanwhile, conservative values of the safety factors against sliding and overturning are

recommended. An alternative option might be that of using a different calculation method, as

described in Section 5.3.

5.2. The Hiroi Formula

The main advantage in using Hiroi’s approach is that calculations become extremely easy, because

the pressure is uniformly distributed along the wall. Figure 16 clearly shows that this formula returns

an upper limit (safe value) of the actual pressures, according to the preliminary indications of

Vicinanza et al. [20]. The magnitude of the overestimation is on average 33%. However, measures and

estimates appear well correlated (R2 is nearly 0.8) and their relationship is clearly linear. This would

indicate that the Hiroi hypothesis of a rectangular pressure distribution is not unrealistic for the present

data set.

Figure 16. Measured wave pressures vs. predictions of the Hiroi formula.

To further investigate this item, the standard deviation of the ratio between predicted and measured

wave pressures (at all the transducers) has been calculated for each test. As shown in Figure 17, the

Hiroi formula gives the minimum standard error for all the experiments; this supports the idea that the

use of a bi-trapezoidal distribution, like that of Goda and Takahashi et al., would not improve the

quality of the estimation respect to a simple rectangular one. On contrary it would worsen it.

0.0

1.0

2.0

3.0

4.0

0.0 1.0 2.0 3.0 4.0

Front

Side

Perfect Agreement

p1/250,meas. [kN/m2]

Hiroi

p1/250,meas. [kN/m2]

Hiroi

p1/250,meas. [kN/m2]

Hiroi

p1/250,meas. [kN/m2]

Hiroi

Energies 2012, 5 4794

Figure 17. Standard deviation of the ratio between measured and calculated p1/250. Left

panel: Front waves; right panel: Side waves.

Hence, based on the Hiroi’s approach, an alternative calculation method could be proposed, which

would have the advantage of simplifying the calculations without any loss of accuracy. Obviously, to

build such a design tool it would be necessary to find a relationship between the intensity of the (mean)

pressure and the main structural and hydraulic variables. A tentative formula is given in the

next paragraph.

5.3. An Alternative Prediction Method

The force acting on the structure can be easily calculated using an approach rather similar to that of

the Japanese practice for sloping top caissons (paragraph 3.2). It is assumed assumed that the bore

which exerts the thrust on the structure is “pushed” forward by the momentum flux, MF, of the wave

field at the toe of the run-up ramp. Since the momentum flux is conserved, Equation (7) still holds with

the sole difference that MF is now derived from the properties of the wave motion at the depth h rather

than from the Goda model. The proposed scheme resembles that used by Hughes [21] for calculating

the run-up height at smooth slopes.

As FP in Equation (7) represents the maximum force over a wave cycle, MF also has to be seen as

the maximum momentum flux in a period T (Hughes, [20]). Hence we have:

max (12)

where u represents the horizontal component of the wave velocity normal to the structure and pd is the

local and instantaneous value of the dynamic pressure associated with the presence of the waves. As

already stated, h is the water depth at the toe of the run-up ramp; η indicates the elevation of the free

surface from the mean sea level.

The simplest way to calculate this momentum flux is by assuming the waves are progressive and

linear. Under this assumption, the convective term in u can be neglected as it is at least of order O(a2)

and consequently much smaller than the diffusive term in p, which is of order O(a). In this way MF

depends only on the contribution of the wave pressure, just like in the Japanese approach. If the sea

bottom is flat one gets:

Energies 2012, 5 4795

2tanh

(13)

in which H is the incident wave height at the toe of the “focuser” and k represents the wave number 2π/L.

For the application to random waves, it is assumed Equation (13) holds wave by wave; hence under

the hypothesis of narrow banded spectrum (H1/250 = 1.8 Hs, T1/250 = Tp) one finally obtains:

, Γ , sin Γ1.82

tanhsin (14)

where the subscript “p” indicates that the wave number is calculated with reference to the peak

wavelength. Γf is an empirical coefficient which compensates for all the approximations introduced

[this explains the symbol ≅ in the Equation (7)]. In particular four aspects deserve to be mentioned:

a. the neglection of wave reflection;

b. the hypothesis of narrow banded waves;

c. the neglection of the phase lag among the pressure transducers when computing the measured

force (Section 4.1)

d. the neglection of the convective term in the expression of MF.

The first three points should not produce a correction factor much different from 1. In particular “a”

and “b” tend to compensate each other. The wave reflection generally enhances the flux of momentum,

as it increases the wave height by (1 + KR2)0.5 on averagely on a wave length (KR is the reflection

coefficient), up to a maximum of (1 + KR) if the structure is located in a pseudo-antinode of the quasi

standing wave field; so in principle a multiplier halfway between the previous expressions would be

needed to correct Equation (13). On the other hand the finite band of the spectrum tends to reduce

MF,250 as H1/250 becomes less than 1.8 Hs. As far as the third point is concerned, the difference between

the real maximum force and the “envelope force” is expected to be quite small, because, as already

stated, the waves are non breaking and accordingly the pressures vary slowly in the time. Thus, a key

role is played by the point “d”; assuming that pd and ρu2 are of the same order of magnitude, it might

be concluded that Γf varies be between 1 and 2. The former value would hold when the hypothesis of

linear waves is correct, the latter if pd is nearly equal to ρu2. In this regard, it could be useful to remark

that theoretically the convective part of the momentum flux might be easily calculated by using either

the Stokes theory or other wave models. In Figure 18, the measured values of FP 250 are compared to , sin [Equation (14)]. The degree

of correlation is excellent and R2 exceeds 0.96. The mean relative error is about 10% and the estimated

value of Γf is about 1.16, which corroborates our previous reasoning. As already mentioned, it is

assumed that FP250 is uniformly distributed (in terms of pressures) along the front face of the structure.

Apart from the arguments discussed in Section 5.2, it should be noted that actually no alternative

distribution law can be deduced from the data. Differently from Hiroi, we make no assumption about

the run-up height so that the wave pressure will be always spread over the entire front wall. This is

basically for two reasons. On the one hand the situations where the device is not overtopped are

generally of no interest for design purposes. On the other hand, whenever they were, no detailed

information is available about the actual run-up levels at SSG.

Energies 2012, 5 4796

Figure 18. FP 250 vs. , sin .

In this frame, the hypothesis of “full” rectangular distribution can be considered acceptable, as it

predicts reasonably well the point of application of the thrust in case of significant overtopping

(according to Hiroi), whereas it is cautious in absence of overpassing, because the baricentre of

loadings is pushed upward, increasing the overturning component of the torque. Figure 19 compares

predicted and measured moments; the agreement is rather satisfactory and the relative error rarely

surpasses 10%.

Before concluding this paragraph, another point deserves to be commented. In the new method, no

correction factor has been introduced to account for the effect of the wave obliquity. This is consistent

with the hypothesis of linearizing the momentum flux, because if it depended primarily on the

pressure, the wave direction would have no influence on the resulting thrust. Actually the experimental

results shown up to now do not indicate any reason to reject this assumption.

Figure 19. Measured vs. predicted overturning moments.

To further investigate this matter, Figure 20 compares the values of the measured pressures (at all

the transducers) for couple of tests having the same wave height and period, but different direction.

y = 1.1609xR² = 0.9633

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6

Data

Fp250 [kN/m]

MF senθ [kN/m]

‐0.14

‐0.12

‐0.10

‐0.08

‐0.06

‐0.04

‐0.02

0.00

‐0.14‐0.12‐0.10‐0.08‐0.06‐0.04‐0.020.00

Front

Side

Perfect Agreement

plus 10% Band

minus 10% Band

M calc [kNm/m]

M meas [kNm/m]

Energies 2012, 5 4797

Although the data are scattered, the graphs do not provide any reasons to conclude that the pressures

exerted by side wave attacks are systematically lower than those exerted by head-on waves. On the

contrary, we note that on the upper plate the values of the pressure are larger for oblique waves;

Vicinanza and Frigaard [6] reasoned this behavior was related to the occurrence of plunging breakers

at the side walls of the structure.

Figure 20. Wave pressures for front and side long crested wave attacks; data refer to

couple of tests having the same wave height and period. Top left panel: Hs = 0.167 m,

Tp = 1.81 s, h = 0.53 m. Top right panel: Hs = 0.167 m, Tp = 1.81 s, h = 0.50 m. Low panel:

Hs = 0.125 m, Tp = 1.55 s, h = 0.50 m.

6. Short Crested Waves

As mentioned in Section 2, the experiments with short crested waves included 16 sea states, eight

with a spreading index n = 4 (large spreading) and eight with n = 10 (small spreading). All the tests

have been run with a water depth h = 0.53 m; each subarray of eight sea states is made up of four front

wave attacks and four side wave attacks. The wave heights and periods are exactly the same as the

corresponding 2D tests, so that the results of the experiments can be easily compared. As shown in

Figure 21, the directional dispersion of the wave energy leads to a weak reduction of the mean

pressure. This phenomenon seems to be independent of the degree of spreading (Figure 22), whereas

the variation of the mean wave direction does not produce any significant effect on the magnitude of

loadings (Figure 23), similarly to what observed for the long crested waves. It is very difficult to

explain this behavior, which may depend on a number of different factors. On the one hand, the

directional spreading might reduce the variance of the wave height distribution in the time domain; this

would imply that, for a given Hs, the extreme waves are smaller than in long crested seas.

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

H=0.167  ‐ h=0.53

p250 F 2D [kN/m2]

p250 S 2D [kN/m2]

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 S 2D [kN/m2]

p250 F 2D [kN/m2]

H=0.167 ‐ h=0.5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0

p250 S 2D [kN/m2]

Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 F 2D [kN/m2]

H=0.125  ‐ h=0.5

Energies 2012, 5 4798

Figure 21. Long crested vs. short crested waves (n = 4, front waves). Top left: Hs = 0.125 m,

Tp = 1.55 s. Top right: Hs = 0.167 m, Tp = 1.81 s. Low left: Hs = 0.25 m, Tp = 2.07 s. Low

right: Hs = 0.125 m, Tp = 1.55 s.

Figure 22. Effect of the spreading index (degree of short-crestedness). Data refer to couple

of tests that differ only by the value of n (n = 10 in abscissas, n = 4 in ordinates). Top

panels (front waves): left: Hs = 0.125 m, Tp = 1.55 s; right: Hs = 0.208 m, Tp = 1.94. Low

panels (side waves): left: Hs = 0.042 m, Tp = 1.03 s; right: Hs = 0.167 m, Tp = 1.81 s.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00

p250 F 3D Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 F 2D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00

p250 F 3D Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 F 2D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

p250 F 3D Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 F 2D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00

p250 F 3D Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 F 2D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00

p250 F 3D n 4 Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 F 3D n 10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00

p250 F 3D n 4 Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 F 3D n 10

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.20 0.40 0.60 0.80 1.00

p250 S 3D n 4 Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 S 3D n 10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00

p250 S 3D n 4 Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

p250 S 3D n 10

Energies 2012, 5 4799

Figure 23. Effect of the mean wave direction on short crested waves. Top left:

Hs = 0.125 m, Tp = 1.55 s, n = 4. Top right: Hs = 0.125 m, Tp = 1.55 s, n = 10. Low left:

Hs = 0.167 m, Tp = 1.81 s, n = 4. Low right: Hs = 0.167 m, Tp = 1.81 s, n = 10.

Figure 24. Measured vs. predicted wave forces (short crested waves).

On the other hand, the phase lag among the Fourier components at the same frequency might reduce

the pressure which really acts on the water column due, for example, to an increase of the vertical

acceleration. At this stage of the research, neither of the above conjectures can be proved and more

experiments and analyses are needed to have a deeper insight on these results. However the reduction

of the mean pressure implies that the methods described in Section 5.1 are generally conservative for

3D waves. As far as the new prediction approach is concerned, a decrease of the scale parameter Γf is

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

H=0.125  ‐ h=0.53

p250 F 3D [kN/m2]

p250 S 3D [kN/m2]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

H=0.125  ‐ h=0.53

p250 F 3D [kN/m2]

p250 S 3D [kN/m2]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

H=0.167  ‐ h=0.53

p250 F 3D [kN/m2]

p250 S 3D [kN/m2]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Lower Plate

Middle Plate

Upper Plate

Perfect Agreement

H=0.167  ‐ h=0.53

p250 F 3D [kN/m2]

p250 S 3D [kN/m2]

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8

Front

Side

Perfect Agreement

plus 10% Band

minus 10% Band

F calc [kN/m]

F meas [kN/m]

Energies 2012, 5 4800

expected. As shown in Figure 24, the mean value of the empirical proportionality factor is now very

close to 1. In the graph, the variables appear still remarkably correlated, being R2 = 0.97. Respect to

the case of long crested waves the prediction of the overturning moments is some less efficient, as

displayed in Figure 25.

Figure 25. Measured vs. predicted wave overturning moments (short crested waves).

6. Summary and Conclusions

In this paper, the random wave experiments of Vicinanza and Frigaard [6] have been re-analyzed, in

order to get possible methods for the estimation of the wave force acting onto the front-face of a

Seawave Slotcone Generator (SSG). Particular attention has been drawn to the Japanese formulae for

the design of monolithic dikes, such as vertical breakwaters, sloping top caissons and trapezoidal

walls. It has been found that considering the system “device + run-up ramp” as a sloping wall, the

well-known Goda formula gives realistic predictions of the total thrust provided that, as suggested by

Tanimoto and Kimura [18], the pressure distribution is rotated from the vertical to the outer face of the

structure. This agrees with preliminary results discussed in [20]. However the model seems to

underpredict the data with low wave height and to overpredict those with high wave height. This

behavior can be corrected by multiplying the pressures by the coefficient [Equation (10)],

originally introduced by Takahashi et al. [16] for sloping top caissons. Yet, with this correction, a

tendency at slightly underestimate data with wave height to depth ratios larger than 0.3 has been

observed; this has to be carefully accounted for in practical applications, when choosing the values of

the safety factors against sliding and overturning.

The comparison with the pioneering Hiroi formula proved the latter may be employed as an upper

limit estimate of the wave force. However, data also showed that the use of a rectangular pressure

distribution, beside simplifying calculations compared to the bi-trapezoidal hypothesis of the Goda

method, produces less scatter between measured and predicted pressures.

In general, the fact that the traditional formulae of the maritime engineering fit only partially the

case of the SSG is not surprising. In this regard one should mind that coastal engineers generally tend

‐0.14

‐0.12

‐0.10

‐0.08

‐0.06

‐0.04

‐0.02

0.00

‐0.14‐0.12‐0.10‐0.08‐0.06‐0.04‐0.020.00

Front

Side

Perfect Agreement

plus 10% Band

minus 10% Band

M calc [kNm/m]

M meas [kNm/m]

Energies 2012, 5 4801

to minimize the wave overtopping occurrence, whilst SSG uses a steep cliff just to enhance it. Then the

design concepts of breakwaters and SSGs must be significantly different.

In the last part of the paper, a new prediction method has been presented, which proved to perform

remarkably well with respect to the present data set. It is based on the momentum flux principle which,

originally proposed by Hughes [21] for the prediction of the run-up height at smooth slopes, resulted

effective in a number of relevant phenomena of the coastal engineering [22]. The degree of correlation

with the data (always larger than 0.95) seems to indicate that the relationships with all the hydraulic

variables are well described. According to the findings discussed above, the estimated wave force has

been assumed to be uniformly distributed along the front plates. In spite of this simplification, good

predictions of the overturning moments have been obtained. This seems to indicate that although the

actual pressure distribution follows a complicated path, its centre of mass lies about halfway up the

structure height. However, new experiments are needed to confirm present results.

As a last but not least remark, it should be noted that all the prediction methods discussed in this

work can be used to calculate the resulting loads (force and moment), but not for estimating the

pressures which locally act onto the front plates. Indeed the latter is important when addressing the

problem of determining the thickness of the single concrete plate and the amount of steel to be used.

This aspect will be specifically handled in a next paper. In the meanwhile, the pressure given by the

Hiroi formula may serve this scope (Figure 18). In this regard it is useful to remark that the maximum

underprediction (unsafe prediction) detected from the data is equal to 14%.

In conclusion, the estimation methods here proposed can be viewed as generally reliable for

practical purposes. This is despite a number of simplified assumptions, among which the fact of

applying results of mostly bi-dimensional model tests (i.e., the Goda and Takahshi formulae) to fully

3D experiments, where the structure width is rather small compared to the extension of the wave front.

In this regard it should be noted this approach is generally conservative as it has long been recognized

that under oblique seas, the lack of wave coherence along the wall produces a reduction of the mean

peak force [23–25]. However this aspect might concern the Japanese empirical formulae, while the

momentum flux approach is supposed to apply to 3D situations as long as the hypothesis of neglecting

the convective term is realistic. On the other hand, the effect of the short crestedness on long structures

appears somewhat less clear (e.g. [26]) and deserves to be studied in depth in the future research.

Acknowledgments

The authors wish to gratefully acknowledge Jens Peter Kofoed as well as an anonymous reviewer

for their fruitful contribution in improving the quality of this paper.

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